We now provide the details of Algorithm 1: Eq. (REF ) denotes the variance of factor matrix \(\mathbf {G}\) , and Eqs. (REF ) and (REF ) denote the variance and mean of vector \(\mathrm {vec}(\mathbf {G})\) , respectively, which is derived in Section IV-C. Eq. (REF ) denotes the mean in the matrix form of factor matrix \(\mathbf {G}\) , which is reshaped from vector \(\mathbf {u}\) . Eqs. (REF ) and (REF ) denote the variance and mean of factor matrix \(\mathbf {X}\) , respectively, which are derived in Section IV-D. Eqs. (REF )-(REF ) denote the hyper-parameters of \(\xi \) , \(\eta \) , \(\gamma \) , and \(\beta \) , respectively, which are derived in Section IV-E.
After applying Algorithm 1, we can obtain \({\lbrace \mathbf {M}_l^{i,t+1}\rbrace _{l=1}^L}_{k=1}^K\)  and \(\mathbf {M}_G^{t+1}\) , which correspond to the estimation of data \(\mathbf {x}_{i,k,l}, \forall i, k, l\) , and the channel sparsity profile \(\lambda _k, \forall k\) . Note that scalar indeterminacy can be resolved by designing each sub-constellation \(\wp _i\)  based on a Grassmannian codebook [1]}.
